How do I UV unwrap a half sphere?
It will be hard to have perfect circles on your ladybug this way. You could do it several other ways, but here are 3 ways to do it:

1) DIRECT PAINTING


Unwrap your semi-sphere in Blender.
Prepare a red image texture in the UV/Image Editor
Plug the Image Texture node with your image in a Diffuse (and plug the Diffuse in the Output).
In the 3D View window, switch to Texture Paint mode and paint your black dots with a very steep curve brush.




2) PAINT A MASK


Unwrap your semi-sphere in Blender.
Create a 100% white texture and plug it in a Diffuse (and plug the Diffuse in the Output).
In the 3D View window, switch to Texture Paint mode and paint your black dots with a very steep curve brush.
Now change your node organization and use your texture as a factor in a mix between 2 Diffuse nodes: one black and one red.




3) TRANSFORM A PLANE INTO A SPHERE

You could also create a plane, map your pattern on it, transform it into a sphere with some Simple Deform modifiers / Bend mode, then cut the sphere in half: How to animate morphing a rectangular plane into a sphere